STUDY ON THE INTERRELATION OF EFFICIENT PORTFOLIOS AND THEIR FRONTIER UNDER t DISTRIBUTION AND VARIOUS RISK MEASURES

In order to study the effect of different risk measures on the efficient portfolios (frontier) while properly describing the characteristic of return distributions in the stock market, it is assumed in this paper that the joint return distribution of risky assets obeys the multivari-ate t-distribution. Under the mean-risk analysis framework, the interrelationship of efficient portfolios (frontier) based on risk measures such as variance, value at risk (VaR), and expected shortfall (ES) is analyzed and compared. It is proved that, when there is no riskless asset in the market, the efficient frontier under VaR or ES is a subset of the mean-variance (MV) efficient frontier, and the efficient portfolios under VaR or ES are also MV efficient; when there exists a riskless asset in the market, a portfolio is MV efficient if and only if it is a VaR or ES efficient portfolio. The obtained results generalize relevant conclusions about investment theory, and can better guide investors to make their investment decision.     

STUDY ON THE INTERRELATION OF EFFICIENT PORTFOLIOS AND THEIR FRONTIER UNDER t DISTRIBUTION AND VARIOUS RISK MEASUREgS

In order to study the effect of different risk measures on the efficient portfolios （fron- tier） while properly describing the characteristic of return distributions in the stock market, it is assumed in this paper that the joint return distribution of risky assets obeys the multivariate t-distribution. Under the mean-risk analysis framework, the interrelationship of efficient portfolios （frontier） based on risk measures such as variance, value at risk （VaR）, and expected shortfall （ES） is analyzed and compared. It is proved that, when there is no riskless asset in the market, the efficient frontier under VaR or ES is a subset of the mean-variance （MV） efficient frontier, and the efficient portfolios under VaR or ES are also MV efficient; when there exists a riskless asset in the market, a portfolio is MV efficient if and only if it is a VaR or ES efficient portfolio. The obtained results generalize relevant conclusions about investment theory, and can better guide investors to make their investment decision.     

Huber＇s Minimax Approach in Distribution Classes with Bounded Variances and Subranges with Applications to Robust Detection of Signals

A brief survey of former and recent results on Huber‘s minimax approach in robust statistics is given. The least informative distributions minimizing Fisher information for location over several distribution classes with upper-bounded variances and subranges are written down. These least informative distributions are qualitatively different from classical Huber‘s solution and have the following common structure: (i) with relatively small variances they are short-tailed, in particular normal；(ii) with relatively large variances they are heavytailed, in particular the Laplace; (iii) they are compromise with relatively moderate variances. These results allow to raise the efficiency of minimax robust procedures retaining high stability as compared to classical Huber‘s procedure for contaminated normal populations. In application to signal detection problems, the proposed minimax detection rule has proved to be robust and close to Huber‘s for heavy-tailed distributions and more efficient than Huber‘s for short-tailed ones both in asymptotics and on finite samples。     

Globally proper saddle point in ic-cone-convexlike set-valued optimization problems

The existence conditions of globally proper efficient points and a useful property of ic- cone-convexlike set-valued maps are obtained. Under the assumption of the ic-cone-convexlikeness, the optimality conditions for globally proper efficient solutions are established in terms of Lagrange multipliers. The new concept of globally proper saddle-point for an appropriate set-valued Lagrange map is introduced and used to characterize the globally proper efficient solutions. The results which are obtained in this paper are proven under the conditions that the ordering cone need not to have a nonempty interior.     

Parametric estimation of discretely sampled Gamma-OU processes

The stationary Gamma-OU processes are recommended to be the volatility of the financial assets. A parametric estimation for the Gamma-OU processes based on the discrete observations is considered in this paper. The estimator of an intensity parameter A and its convergence result are given, and the simulations show that the estimation is quite accurate. Assuming that the parameter A is estimated, the maximum likelihood estimation of shape parameter c and scale parameter a, whose likelihood function is not explicitly computable, is considered. By means of the Gaver-Stehfest algorithm, we construct an explicit sequence of approximations to the likelihood function and show that it converges the true (but unkown) one. Maximizing the sequence results in an estimator that converges to the true maximum likelihood estimator and the approximation shares the asymptotic properties of the true maximum likelihood estimator. Some simulation experiments reveal that this method is still quite accurate in most of rational situations for the background of volatility.     

Uniform Convergence Rate of Estimators of Autocovariances in Partly Linear Regression Models with Correlated Errors

Consider the partly linear regression model ,where yi's are responses, xi = (xi1, xi2,…,xip)' and ti ∈T are known and nonrandom design points, T is a compact set in the real line is an unknown parameter vector, g(·) is an unknown function and {Ei} isa linear process, i.e., random variables with zeromean and variance o2e. Drawing upon B-spline estimation of g(·) and least squares estimation of 0, we construct estimators of the autocovariances of {Ei}- The uniform strong convergence rate of these estimators to their true values is then established. These results not only are a compensation for those of [23], but also have some application in modeling error structure. When the errors {Ei} are an ARMA process, our result can be used to develop a consistent procedure for determining the order of the ARMA process and identifying the non-zero coefficients of the process. Moreover, our result can be used to construct the asymptotically efficient estimators for parameters in the ARMA error process.     

同时序约束下保序正态均值的均方损失

For two normal populations with unknown means μ and unknown variances σ2, assume that there are simple order restrictions among the means and variances: μ1 < μ2 and σ12 >σ22 > 0. This case is said to be simultaneous order restriction by Shi (Maximum likelihood estimation of means and variances from normal populations under simultaneous order restrictions, J. Multivariate Anal., 50(1994), 282-293.) and an iterative algorithm of computing the order restricted maximum likelihood estimates of μi and σi2 was given in that paper. This paper shows that the restricted maximum likelihood estimate of μi has smaller mean square loss than the usual estimate xi under some conditions.     

Robust estimation in inverse problems via quantile coupling

In this article we consider a sequence of hierarchical space model of inverse problems.The underlying function is estimated from indirect observations over a variety of error distributions including those that are heavy-tailed and may not even possess variances or means.The main contribution of this paper is that we establish some oracle inequalities for the inverse problems by using quantile coupling technique that gives a tight bound for the quantile coupling between an arbitrary sample p-quantile and a normal variable,and an automatic selection principle for the nonrandom filters.This leads to the data-driven choice of weights.We also give an algorithm for its implementation.The quantile coupling inequality developed in this paper is of independent interest,because it includes the median coupling inequality in literature as a special case.     

Efficient estimation of varying coefficient seemly unrelated regression model

In this paper, we propose a class of varying coefficient seemingly unrelated regression models, in which the errors are correlated across the equations. By applying the series approximation and taking the contemporaneous correlations into account, we propose an efficient generalized least squares series estimation for the unknown coefficient functions. The consistency and asymptotic normality of the resulting estimators are established. In comparison with the ordinary/east squares ones, the proposed estimators are more efficient with smaller asymptotical variances. Some simulgtlon＇studies and a real application are presented to demonstrate the finite sample performance of the proposed methods. In addition, based on a B-spline approximation, we deduce the asymptotic bias and variance of the proposed estimators.     

GENERALIZED LEAST SQUARES AND MAXIMUM LIKELIHOOD ESTIMATIONS OF MULTIVARIATE POLYCHORIC CORRELATIONS

This paper investigates the generalized least squares estimation and the maximum likelihoodestimation of the parameters in a multivariate polychoric correlations model,based on data from amultidimensional contingency table.Asymptotic properties of the estimators are discussed.An iterativeprocedure based on the Gauss-Newton algorithm is implemented to produce the generalized leastsquares estimates and the standard errors estimates.It is shown that via an iteratively reweightedmethod,the algorithm produces the maximum likelihood estimates as well.Numerical results on thefinite sample behaviors of the methods are reported.     

Timber harvest scheduling with price uncertainty using Markowitz portfolio optimization

Harvest scheduling models need to account for uncertain revenue predictions when minimizing risk of financial loss is an important management objective. In this paper, we present methods for estimating the means and covariances of stumpage prices and incorporating them in harvest scheduling models. We approached the estimation problem by fitting time-series models to loblolly pine sawtimber and pulpwood stumpage prices in Georgia, USA, and deriving formulas for means and covariances of price predictions. Statistical evidence supported integrated autoregressive models, which caused covariances of price predictions to increase with time. The means and covariances of price predictions were combined with timber yield and land value predictions to give exact formulas for the revenue means and covariances of timber management activities. Sawtimber regimes dominated pulpwood regimes by providing higher mean revenues across a wide range of revenue variances. Harvest scheduling results for a hypothetical forest of pine plantations showed that the forest plan that maximized mean income without concern for risk (expressed as the standard deviation of income) involved sawtimber production with a 35-year rotation age. Risk was reduced 30% with little effect on mean income by using shorter-rotation sawtimber regimes. Risk was reduced 80% by using a mix of short-rotation sawtimber and pulpwood regimes because pulpwood price was only weakly correlated with sawtimber price. The latter risk-reduction came at the expense of mean income, which was reduced by as much as 50%. The risks and compositions of optimal forest plans were extremely sensitive to assumptions about the range of future prices that were inherent in different prediction models. This sensitivity emphasizes the importance of carefully determining the decision makers beliefs about stumpage price behavior.     

Maximum-likelihood adjoint-state finite-element estimation of groundwater parameters under steady- and nonsteady-state conditions

A method is presented for estimating the hydraulic parameters of groundwater flow models under steady- and nonsteady-state conditions. The estimation problem is posed in the framework of maximum-likelihood theory by means of a log-likelihood criterion that includes prior estimates of the parameters. To allow for an incomplete knowledge of the covariances of the prior head and parameter errors, these covariances are expressed in terms of a few unknown statistical parameters that may be estimated jointly with the hydraulic parameters. Computational efficiency is achieved by evaluating the gradient of the estimation criterion with an adjoint-state finite-element scheme and using a combination of conjugate-gradient algorithms, coupled with Newton's method for determining the step size to be taken at each iteration. Model structure identification criteria developed in the time-series literature (all of which utilize the maximum-likelihood concept) are shown to be useful for selecting the best way to parametrize a groundwater flow region when a number of alternative schemes of parametrization are given. The paper also demonstrates the potential utility of the proposed estimation method for the optimum design of space-time measurement networks. A case study dealing with three-dimensional flow in a multiaquifer system is briefly discussed.     

Obtaining distributional information from valuation lattices

Efficient algorithms for obtaining information about the total return distribution of securities from valuation lattices are described. This information, including variances and covariances between securities, is useful when constructing hedging transactions that achieve specific objectives.     

Minimum Disparity Estimation in Linear Regression Models: Distribution and Efficiency

This paper deals with the minimum disparity estimation in linear regression models. The estimators are defined as statistical quantities which minimize the blended weight Hellinger distance between a weighted kernel density estimator of errors and a smoothed model density of errors. It is shown that the estimators of the regression parameters are asymptotic normally distributed and efficient at the model if the weights of the density estimators are appropriately chosen.     

Asymptotic Joint Normality of Outdegrees of Nodes in Random Recursive Trees

We study the joint probability distribution of the number of nodes of outdegree 0, 1, and 2 in a random recursive tree. We complete the known partial list of exact means and variances for outdegrees up to two by obtaining exact combinatorial expressions for the remaining means, variances, and covariances. The joint probability distribution of the number of nodes of outdegree 0, 1, and 2 is shown to be asymptotically trivariate normal and the asymptotic covariance structure is explicitly determined. It is also shown how to extend the results (at least in principle) to obtain a limiting multivariate normal distribution for nodes of outdegree 0, 1, …, k.     

UMVU estimation of the ratio of powers of normal generalized variances under correlation

We consider estimation of the ratio of arbitrary powers of two normal generalized variances based on two correlated random samples. First, the result of Iliopoulos [Decision theoretic estimation of the ratio of variances in a bivariate normal distribution, Ann. Inst. Statist. Math. 53 (2001) 436-446] on UMVU estimation of the ratio of variances in a bivariate normal distribution is extended to the case of the ratio of any powers of the two variances. Motivated by these estimators’ forms we derive the UMVU estimator in the multivariate case. We show that it is proportional to the ratio of the corresponding powers of the two sample generalized variances multiplied by a function of the sample canonical correlations. The mean squared errors of the derived UMVU estimator and the maximum likelihood estimator are compared via simulation for some special cases.     

A-optimal chemical balance weighing design with correlated errors

In this paper we study the estimation problem of individual weights of objects using an A-optimal chemical balance weighing design. We assume that in this model errors are correlated and they have the same variances. The lower bound oftr (X′G ^?1 X)^?1 is obtained and a necessary and sufficient condition for this lower bound to be attained is given. There is given new construction method of A-optimal chemical balance weighing design.     

Estimators of covariances in time series models

The theory of Minimum Norm Quadratic Estimators for estimating variances and covariances is applied to show that some commonly used estimators of covariances in time series models are easily derived using the above principle. In applying the theory MINQE, it is observed that no unbiased estimator exists in the class of invariant quadratics.     

A stochastic population projection system based on general age-dependent branching processes

"Algorithms for a stochastic population process, based on assumptions underlying general age-dependent branching processes in discrete time with time inhomogeneous laws of evolution, are developed through the use of a new representation of basic random functions involving birth cohorts and random sums of random variables. New algorithms provide a capability for computing the mean age structure of the process as well as variances and covariances, measuring variation about means. Four exploratory population projections, testing the implications of the algorithms for the case of time-homogeneous laws of evolution, are presented. Formulas extending mean and variance functions for unit population projections...are also presented. These formulas show that, in population processes with non-random laws of evolution, stochastic fluctuations about the mean function are negligible when initial population size is large. Further extensions of these formulas to the case of randomized laws of evolution suggest that stochastic fluctuations about the mean function can be significant even for large initial populations."